Since the standard symmetrized variance \(\widehat{V}(|\psi \rangle )\)  is
a real symmetric concave function of the coherence vector \(\left(r_0,r_1, \ldots , r_{d-1}\right)^\) , and is zero if and only if \(|\psi \rangle \) 
is incoherent, \(\widehat{V}_c(\rho )\)  is a coherence measure , [1]}. The convex roof extension \(\widehat{V}_c(\rho )\) 
satisfies the following properties. (i) \(\widehat{V}_c(\rho )\)  is
convex; (ii) \(0\leqslant \widehat{V}_c(\rho )\leqslant 1-\frac{1}{d}\) . Moreover, \(\widehat{V}_c(\rho )=0\)  if and only if
\(\rho \)  is incoherent, and \(\widehat{V}_c(\rho )=1-\frac{1}{d}\)  if and
only if \(\rho \)  is maximally coherent; (iii) For \(\rho =p_1 \rho _1\oplus p_2 \rho _2\) , one has \(\widehat{V}_c(p_1 \rho _1 \oplus p_2\rho _2)={p_1}\widehat{V}_c(\rho _1) + {p_2} \widehat{V}_c(\rho _2)\)  [2]}.
